They had a chance to tie it up in the final minute, but Matt Ryan's fourth-down pass went off the hands of Roddy White in the end zone with 36 seconds remaining as Aqib Talib defended tightly on the play.
New England seemed in control when LeGarrette Blount scored on a 47-yard run, and Brady threw an 18-yard touchdown pass to rookie Kenbrell Thompkins.
Thompkins finished with six catches for 127 yards nearly as much as the first three games of his career. Julian Edelman chipped in with 118 yards on seven receptions.
Atlanta (1-3) dropped two games below .500 for the first time since 2007 despite a career night from Tony Gonzalez. He had 12 catches for 149 yards and a pair of touchdowns.
New England led just 13-10 before a wild final period. Blount stunned the Georgia Dome crowd with his burst through a gaping hole, cutting outside and dragging defenders into the end zone.
After Matt Bryant's 45-yard field goal brought the Falcons to 20-13, Brady guided the Patriots on an 80-yard touchdown rive that took only five plays and barely two minutes.
Edelman hauled in a 16-yard pass, followed by 44-yard reception that set up the touchdown throw to Thompkins in the left corner of the end zone, sending many of the Atlanta fans heading to the exits with nearly 10 minutes left.
They nearly missed an epic comeback after Stephen Gostkowski's third field goal pushed the Patriots to a 17-point lead.
Gonzalez hauled in an 11-yard touchdown pass, the Falcons recovered an onside kick, then drove for another field goal by Bryant.
New England recovered a second onside kick, but fumbled a snap trying to convert on fourth down, turning it back over to the Falcons.
Ryan completed a 49-yard pass to Julio Jones to put Atlanta in position to force overtime before a mostly empty stadium.